Why are my objects flattened or distorted when using the Shrinkwrap modifier?
The shrinkwrap modifier moves vertices against the target object. The effect you are seeing is the intended effect, as it has no notion of maintaining the volume of an object.

To move the whole object, while maintaining the shape of the object, you can use a shrinkwrap constraint, which will move the object as a whole to place it's origin onto the target surface.